The Siluro-Devonian adaptive radiation of gnathostomes, which underpins almost all living vertebrate biodiversity, is characterised by the evolutionary innovation of the jaw. Previous work revealed stasis in jaw mechanics through the Devonian, following an initial burst in the late Silurian. Here we focus on the constraints on jaw morphology through the investigation of the form and functional performance of jaws across theoretical morphospace. This was achieved by generating a grid of theoretical shapes which encapsulates the range of early gnathostome jaw shape using Elliptical Fourier Analysis (EFA). The functional optimality of each shape was assessed based on resistance to stress and a metric of rotational efficiency, which were combined using a novel Pareto ranking framework. The resultant adaptive landscape was used to interrogate the patterns of jaw morphospace exploration in early gnathostomes. We find that the mandibles of early jawed vertebrates exhibit increasing morphological variance over time contrasting with static functional variety. Gnathostome jaws evolved from initial morphologies that were already optimised for stress resistance and speed of jaw closure, criteria incompatible with jaws evolving first to enhance ventilation. Later jawed vertebrate lineages expand into less optimal regions of shape space, possibly because of additional conflicting selection pressures, but most jaws are Pareto optimal. Functional constraints on early jaw evolution may have become more flexible through time.